The space underlying the projective unitary group of a separable, infinite-dimensional Hilbert space has a number of topologies, so for the purposes of this question, pick you favourite and answer for that one.

I've read that $PU(H)$ is a Fréchet manifold, but that was without saying which topology. There are two ways I can see to think about this. First is that if we know what sort of manifold $U(H)$ is, then we know what $PU(H)$ is, as the former looks locally like a chart of $PU(H)$ times $U(1)$. So we can consider the closed (I think!) subspace $U(H) \subset End(H)$.

Alternately we can consider $PU(H)$ as sitting inside the Hilbert-Schmidt operators on $H$ as the projective unitaries act freely on the latter. (EDIT: this is not right, as pointed out by Andrew. I was thinking of the inclusion $PU(H) \hookrightarrow U(HS(H))$, which doesn't really tell us much in hindsight.)

I'm not familiar enough with the analysis to turn the above observations into results, and I may be interested in other topologies.

So the question is, is there the structure of a Banach or even Hilbert manifold on $PU(H)$?